50 



Major P. A. MacMahon 



[Feb. 14, 



WEEKLY EVENING MEETING, 



Friday, February U, 1902. 



Sir William de VV. Abxey, K.C.B. D.C.L. F.K.S., 

 Vice-President, in the Chair. 



Major P. A. MacMahon, R.A. D.Sc. F.R.S. 



Magic Squares and other ProUems on a Chess-Board. 



The construction of magic squares is an amusement of great an- 

 tiquity ; we hear of them being constructed in India and in China 

 before the Christian era, whilst they appear to have been introduced 

 into Europe by Moschopulus, who flourished at Constantinople early 

 in the fifteenth century. On the diagram you see a simple example 

 of a magic square, one celebrated as being drawn by Albert Diirer in 

 his picture of " Melancholy," painted about the year 1500 (Fig. 1). 



It is one of the fourth order, involving 

 16 compartments or cells. In describing 

 such squares, the horizontal lines of 

 cells are called " rows," the vertical lines 

 " columns," and the oblique lines going 

 from corner to corner *' diagonals." In 

 the 16 compartments are placed the first 

 16 numbers, 1, 2, 3, . . . 16, and the 

 magic property consists in this, that the 

 numbers are placed in such wise that 

 the sum of the numbers in every row, 

 column, and diagonal is the same, viz. in 

 this case, 34. 



It is probable that magic squares 

 were so called because the properties 

 they possessed seemed to be extraordinary and wonderful ; they 

 were, indeed, regarded with superstitious reverence, and em23loyed 

 as talismans. Cornelius Agrippa constructed magic squares of orders 

 3, 4, 5, 6, 7, 8, 9, and associated them with the seven heavenly 

 bodies, Saturn, Jupiter, Mars, the Sud, Venus, Mercury, and the 

 Moon. A magic square engraved on a silver plate was regarded as 

 a charm against the plague, and to this day such charms are worn 

 in the East. 



However, what was at first merely a practice of magicians and 

 talisman makers has now for a long time become a serious study for 

 mathematicians. Not that they have imagined that it would lead 

 them to anything of solid advantage, but because the theory of such 



Fig. 1. 



